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Let $X$ be a smooth projective complex variety and $\mathcal{F}$ a coherent sheaf on $X$.

Let $\tau$ be a Grothendieck topology and $$ \chi(X,\mathcal{F},\tau)=\sum_i(-1)^i dim_{\mathbb{C}}H^i_\tau(X,\mathcal{F}) $$ the Euler characteristic of $(X,\mathcal{F},\tau)$.

For a compact closed and connected manifold $\bar X$, the Euler characteristic is $$ \chi(\bar X)=\sum_i(-1)^i rank_{\mathbb{Z}}H^i(\bar X,\mathbb{Z}). $$ Note that we may also use singular homology instead of singular cohomology. The number $rank_{\mathbb{Z}}H^i(\bar X,\mathbb{Z})$ is the number of $\mathbb{Z}$-factors of the group $H^i(\bar X,\mathbb{Z})$, the $i$-th Betti number of $X$. This number equals $dim_{\mathbb{C}}(H^i(\bar X,\mathbb{Z})\otimes\mathbb{C})$. This should be the same as $dim_{\mathbb{C}}H^i(\bar X,\mathbb{C})$ if I am not mistaken.

To the variety $X$ we can associate its analytification $\bar X$.

How do $\chi(X,\mathcal{F},\tau)$ and $\chi(\bar X)$ relate?

This question rises from this one where it is said that there is no nice relation for $\tau$ the Zariski topology. In the answers however, a connection with the etale topology and $F=\mathcal{O}_X$ the structure sheaf is suggested. What is this connection precisely? I remember vaguely that one should take finite coefficients somewhere, but I don't see where and why. In this case there is no such direct connection between singular homology and cohomology which makes the thing even more difficult for me to understand.

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